

A160009


Numbers that are the product of distinct Fibonacci numbers.


25



0, 1, 2, 3, 5, 6, 8, 10, 13, 15, 16, 21, 24, 26, 30, 34, 39, 40, 42, 48, 55, 63, 65, 68, 78, 80, 89, 102, 104, 105, 110, 120, 126, 130, 144, 165, 168, 170, 178, 195, 204, 208, 210, 233, 240, 267, 272, 273, 275, 288, 312, 315, 330, 336, 340, 377, 390, 432, 440, 442, 445
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OFFSET

1,3


COMMENTS

Starts the same as A049862, the product of two distinct Fibonacci numbers. This sequence has an infinite number of consecutive terms that are consecutive numbers (such as 15 and 16) because fib(k)*fib(k+3) and fib(k+1)*fib(k+2) differ by one for all k >= 0.
It follows from Carmichael's theorem that if u and v are finite sets of Fibonacci numbers such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. The same holds for many other 2nd order linear recurrence sequences with constant coefficients. In the following guide to related "distinct product sequences", W = Wythoff array, A035513:
base sequence distinctproduct sequence
A000045 (Fibonacci) A160009
A000032 (Lucas, without 2) A274280
A000032 (Lucas, with 2) A274281
A000285 (1,4,5,...) A274282
A022095 (1,5,6,...) A274283
A006355 (2,4,6,...) A274284
A013655 (2,5,7,...) A274285
A022086 (3,6,9,...) A274191
row 2 of W: (4,7,11,...) A274286
row 3 of W: (6,10,16,...) A274287
row 4 of W: (9,15,24,...) A274288
 Clark Kimberling, Jun 17 2016


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000


MATHEMATICA

s={1}; nn=30; f=Fibonacci[2+Range[nn]]; Do[s=Union[s, Select[s*f[[i]], #<=f[[nn]]&]], {i, nn}]; s=Prepend[s, 0]


CROSSREFS

A059844, A065108
Sequence in context: A177445 A022826 A053035 * A049862 A022829 A229172
Adjacent sequences: A160006 A160007 A160008 * A160010 A160011 A160012


KEYWORD

nonn


AUTHOR

T. D. Noe, Apr 29 2009


STATUS

approved



